Math 295 Spring Term 2008
Assignment 14
1. (Pohoˇzaev’s identity) Assume that g∈C(R,R), G(u) = Ru
0g(v)dv
and that Ω ⊂Rnis bounded with smooth boundary. Let ube a
classical solution of
(−4u=g(u) in Ω
u= 0 on ∂Ω
and show that it satisfies
nZΩ
G(u)dx +2−n
2ZΩ
u g(u)dx =1
2Z∂Ω
(∇u·ν)2(x·ν)dσ
[Hint: Use Gauss theorem with the vector field V(x) = (x· ∇u)∇u.]
2. Use Pohoˇzaev’s identity to prove that no nontrivial solution can exist
for (−4u=|u|pin Ω
u= 0 on ∂Ω
if p > n+2
n−2and Ω is a star-shaped bounded Lipschitz domain in Rn.
3. Let Xbe a normed vector space. Prove that a convex functional
φ:X→Ris continuous at x∈Xif it is bounded in a neighbor-
hood of x. Give an example of a convex functional which is nowhere
continuous.
4. Let β∈C∞(R) satisfying β0(R)⊂[δ, σ] for δ , σ ∈(0,∞). Give a
weak formulation of
(−4u=fin Ω
∂νu+β(u) = 0 on Ω .
in an open bounded domain Ω ⊂Rnand prove that it possesses a
weak solution.
5. For an open and bounded Ω ⊂Rnlet
A=u∈H1
0(Ω,Rm)u=gon ∂Ω,|u|= 1 a.e..
Show that φdefined by φ(u) = 1
2RΩ|Du(x)|2dx has at least one
minimizer in A(if A 6=∅) and that any minimizer satisfies
ZΩ
Du(x) : Dv(x)dx =ZΩ
|Du(x)|2u(x)v(x)dx ,
v∈H1
0(Ω,Rm)∩L∞(Ω,Rm).
Homework due by Wednesday, May 21 2008